Modular equalities for complex reflection arrangements

TL;DR

This paper computes the combinatorial Aomoto-Betti numbers for complex reflection arrangements, revealing bounds, specific conditions for non-zero values, and their relation to monodromy eigenvalues, with applications to full monomial arrangements.

Contribution

It provides a uniform combinatorial characterization of eigenvalue multiplicities and computes Aomoto-Betti numbers for complex reflection arrangements, including full monomial cases.

Findings

eta_p( ext{A}) ext{ is at most } 2 for rank ≥ 3 arrangements.

eta_p( ext{A})=0 ext{ for primes } p>3.

e_d( ext{A}) ext{ equals } eta_p( ext{A}) ext{ for prime } d.

Abstract

We compute the combinatorial Aomoto-Betti numbers $\beta_p(\mathcal{A})$ of a complex reflection arrangement. When $\mathcal{A}$ has rank at least $3$, we find that $\beta_p(\mathcal{A})\le 2$, for all primes $p$. Moreover, $\beta_p(\mathcal{A})=0$ if $p>3$, and $\beta_2(\mathcal{A})\ne 0$ if and only if $\mathcal{A}$ is the Hesse arrangement. We deduce that the multiplicity $e_d(\mathcal{A})$ of an order $d$ eigenvalue of the monodromy action on the first rational homology of the Milnor fiber is equal to the corresponding Aomoto-Betti number, when $d$ is prime. We give a uniform combinatorial characterization of the property $e_d(\mathcal{A})\ne 0$, for $2\le d\le 4$. We completely describe the monodromy action for full monomial arrangements of rank $3$ and $4$. We relate $e_d(\mathcal{A})$ and $\beta_p(\mathcal{A})$ to multinets, on an arbitrary arrangement.